New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a...

A lower semicontinuity result in BV is obtained for quasiconvex integrals with subquadratic growth. The key steps in this proof involve obtaining boundedness properties for an extension operator, and a precise blow-up technique that uses fine properties of Sobolev maps. A similar result is obtained by Kristensen in [Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], where there are weaker asssumptions on convergence but the integral needs to satisfy a stronger growth condition.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with $\mu$. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu$. More precisely, for fully general $\mu$, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu$, turns out to be necessary for lower...

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where μ is a given positive measure on ${ℝ}^n$, and the vector-valued function u belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower...

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case ${}^{\text{'}}(ℝⁿ,ℝ){\to }^{\nabla }\text{'}(ℝⁿ,ℝⁿ){\to }^{curl}\text{'}(ℝⁿ,{ℝ}^{n\times n})$.

The lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $\left(W^{1,1}\times L^p\right)$-weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope is also provided.

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting...

We present Z. Naniewicz method of optimization a coercive integral functional 𝒥 with integrand being a minimum of quasiconvex functions. This method is applied to the minimization of functional with integrand expressed as a minimum of two quadratic functions. This is done by approximating the original nonconvex problem by appropriate convex ones.